# Define $f : \mathbb{Z}_p \to \mathbb{Z}_p$ by $f([x])=[ax],a\in\mathbb{Z},p\nmid a$. Prove that $f$ is 1-to-1 & onto

Let $$p$$ be prime and define $$f : \mathbb{Z}_p \to \mathbb{Z}_p$$ by $$f([x])=[ax],a\in\mathbb{Z},p\nmid a$$. Prove that $$f$$ is 1-to-1 and onto.

The question is equivalent to proving that $$f$$ permutes the elements of $$\mathbb{Z}_p$$. I know that proving that $$f$$ is invertible will prove that $$f$$ is one-to-one, but I'm not sure how to do this. Also, I know I'll have to use some properties of congruences modulo primes.

• Since the set is finite, it is enough to prove 1-to-1. If $f([x])=f([y])$, then $[ax]=[ay]$. This means that $a(x-y)$ is divisible by $p$. Since $p$ is prime and doesn't divide $a$ it follows that $p$ divides $x-y$. This means that $[x]=[y]$. – conditionalMethod Oct 20 at 4:20

If $$f([x]) = f([y])$$ then $$[ax] = [ay]$$, thus $$p|(ax-ay) \implies p|a(x-y) \implies p|(x-y) \implies [x] = [y]$$ then $$f$$ is $$1-1$$
Now let $$[x] \in \mathbb{Z}_{p}$$
We have that $$(p,a) = 1$$ then there exists $$s,t \in \mathbb{Z}$$ such that $$1 = as + pt$$ thus $$p|(1-as)$$ $$\implies [as] = [1] \implies [x] = [asx] = f([sx])$$ then $$f$$ is onto
So $$\operatorname {gcd}(a,p)=1$$. Thus there exist $$b,c$$ such that $$ba+cp=1$$. That is $$b\cong a^{-1}\pmod p$$. So $$f^{-1}([x])=[bx]$$. You can check that $$f^{-1}$$ is a well-defined inverse for $$f$$. Thus $$f$$ is invertible, hence bijective.